22 Concepts in a Probabilistic Language of Thought

Transcription

1 22 Concepts in a Probabilistic Language of Thought Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg 22.1 Introduction Knowledge organizes our understanding of the world, determining what we expect given what we have already seen. Our predictive representations have two key properties: they are productive, and they are graded. Productive generalization is possible because our knowledge decomposes into concepts elements of knowledge that are combined and recombined to describe particular situations. Gradedness is the observable effect of accounting for uncertainty our knowledge encodes degrees of belief that lead to graded probabilistic predictions. To put this a different way, concepts form a combinatorial system that enables description of many different situations; each such situation specifies a distribution over what we expect to see in the world, given what we have seen. We may think of this system as a probabilistic language of thought (PLoT), in which representations are built from language-like composition of concepts, and the content of those representations is a probability distribution on world states. The purpose of this chapter is to formalize these ideas in computational terms, to illustrate key properties of the PLoT approach with a concrete example, and to draw connections with other views of conceptual structure. People are remarkably flexible at understanding new situations, guessing at unobserved properties or events, and making predictions on the basis of sparse evidence combined with general background knowledge. Consider the game of tug-of-war: two teams matching their strength by pulling on either side of a rope. If a team containing the first author (NG) loses to a team containing the third author (TG), that might provide weak evidence that TG is the stronger of the two. If these teams contain only two members each, we might believe more in TG s greater strength than if the teams contain eight members each. If TG beats NG in a one-on-one tug-of-war, and NG goes on to beat three other individuals in similar one-on-one contests, we might believe that TG is not only stronger than NG but strong in an absolute sense, relative to the general population, even though we have only directly observed TG participating in a single match. However, if we later found out that NG did not try very hard in his match

2 624 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg against TG, but did try hard in his later matches, our convictions about TG s strength might subside. This reasoning is clearly statistical. We may make good guesses about these propositions, but we will be far from certain; we will be more certain of TG s strength after seeing him play many games. But our reasoning is also highly abstract. It is not limited to a particular set of tug-of-war contestants. We can reason in analogous ways about matches between teams of arbitrary sizes and composition, and are unperturbed if a new player is introduced. We can also reason about the teams as collections: if team Alpha wins its first four matches but then loses to team Bravo, whom it has not faced before, we judge team Bravo very likely to be stronger than average. The smaller team Bravo is, the more likely we are to judge a particular member of team Bravo to be stronger than the average individual. And similar patterns of reasoning apply to inferences about skill and success in other kinds of team contests: We could be talking about math teams, or teams for doubles Ping-Pong, and make analogous inferences for all the situations above. Our reasoning also supports inferences from complex combinations of evidence to complex conclusions. For example, suppose that participants have been paired up into teams of two. If we learn that NG was lazy (not trying hard) whenever his team contested a match against TG s team, but NG s team nonetheless won each of these matches, it suggests both that NG is stronger than TG and that NG is often lazy. If we then learned that NG s teammate is stronger than any other individual in the population, we would probably revise the former belief (about NG s strength) but not the latter (about his laziness). If we learned that NG s team had won all of its two-on-two matches, but we were told nothing about NG s teammate, it is a good bet that the teammate whoever he is is stronger than average; all the more so if we also learned that NG had lost several one-on-one matches while trying hard. Finally, our reasoning in this one domain can be modularly combined with knowledge of other domains, or manipulated based on subtle details of domain knowledge. If we observed TG lifting a number of very heavy boxes with apparent ease, we might reasonably expect his tug-of-war team to beat most others. But this would probably not raise our confidence that TG s math team (or even his Ping-Pong team) are likely to be unusually successful. If we know that NG is trying to ingratiate himself to TG, perhaps to receive a favor, then we might not weight his loss very heavily in estimating strength. Likewise if we knew that NG had received a distracting text message during the match. We will return to this extended example in more detail in section 22.3, but for now we take it merely as an illustration, in one simple domain, of the key features of human cognition that we seek to capture in a general computational architecture. How can we account for the wide range of flexible inferences people draw from diverse patterns of evidence such as these? What assumptions about the cognitive system are needed to explain the productivity and gradedness of these inferences? What kind of representations are abstract enough to extend flexibly to novel situations and questions,

3 Concepts in a Probabilistic Language of Thought 625 yet concrete enough to support detailed quantitative predictions about the world? There are two traditional, and traditionally opposed, ways of modeling reasoning in higher-level cognition, each with its well-known strengths and limitations. Symbolic approaches (e.g., Newell, Shaw, and Simon 1958) can naturally formulate a wide array of inferences but are traditionally confined to the realm of certainty. They would be challenged to capture all the gradations of reasoning people find so intuitive and valuable in an uncertain world. Probabilistic network approaches whether based on neural networks (Rumelhart and McClelland 1988) or Bayesian networks (Pearl 1988, 2000) support graded inferences based on sparse or noisy evidence, but only over a fixed finite set of random variables. They lack the representational power and productivity of more structured symbolic approaches and would be hard pressed to formulate in a coherent fashion all the inferences described above let alone the infinite number of similar inferences we could have listed but did not. More recently, researchers have begun to move beyond the dichotomy between statistical and symbolic models (Anderson 1996) and have argued that much of cognition can be understood as probabilistic inference over richly structured representations (Tenenbaum et al. 2011). This has led to a proliferation of structured statistical models, which have given real insight into many cognitive domains: inductive reasoning (Kemp and Tenenbaum 2009), causal learning (Griffiths and Tenenbaum 2009; Goodman, Ullman, and Tenenbaum 2011), word learning (Frank, Goodman, and Tenenbaum 2009; Piantadosi, Tenenbaum, and Goodman 2012), and mental state inference (Baker, Saxe, and Tenenbaum 2009), to name a few. But the computational tools used for these different models do not yet amount to a truly general-purpose or integrated approach. They are both insufficiently flexible requiring new models to be described for each different situation and idiosyncratic requiring different representational tools across different specific domains. We require a new computational architecture for cognition, grounded in a new theory of concepts a theory that does justice to two distinct but equally important roles that concepts play in mental life. On the one hand, concepts enable predictive generalization: they summarize stable regularities, such as typical distributions of objects, properties, and events. This is the role primarily addressed by prototype (Rosch 1999), exemplar (Nosofsky 1986) and other statistical accounts of concepts. On the other hand, concepts provide the basic building blocks of compositional thought: they can be flexibly combined with other concepts to form an infinite array of thoughts in order to reason productively about an infinity of situations. They can be composed to make new concepts, which are building blocks of yet more complex thoughts. Indeed, concepts get much of their meaning and their function from the role they play in these larger-scale systems of thought. These are the roles primarily addressed (albeit it in different ways) by classical (rule-based) theories of concepts (Bruner, Goodnow, and Austin 1967), and by the theory theory (Gopnik 2003) and other accounts based on inferential or conceptual roles (Block 1997). While these theories differ in crucial

4 626 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg ways, we group them under the heading of symbolic approaches, because they highlight the compositional aspects of concepts that require a powerful symbol-processing architecture. Our goal in this chapter is to sketch a new account of concepts that combines these two aspects, their statistical and symbolic functions, and to show how this account can explain more of the richness of human reasoning than has been previously captured using traditional approaches. We can phrase our hypothesis, somewhat informally, as follows: Probabilistic language of thought hypothesis (informal version): Concepts have a language-like compositionality and encode probabilistic knowledge. These features allow them to be extended productively to new situations and support flexible reasoning and learning by probabilistic inference. This view of the nature of concepts provides a deeper marriage of statistical inference and symbolic composition. Because they are probabilistic, concepts support graded reasoning under uncertainty. Because they are language-like, they may be flexibly recombined to productively describe new situations. For instance, we have a set of concepts, such as strength and game, for the tug-of-war reasoning domain described above that we may compose with each other and with symbols referring to entities (individuals and teams) in the domain. These combinations then describe distributions on possible world states, which we may reason about via the rules of probability. Our proposal for the PLoT can be seen as making the statistical view of concepts more flexible and systematic by enriching it with a fine-grained notion of composition coming from symbolic approaches. It can also be seen as making symbolic approaches to concepts more useful for reasoning in an uncertain world, by embedding them in a probabilistic framework for inference and decision. The level of description intended in the PLoT hypothesis is neither the highest level, of input-output relations, nor the lower level of psychological processing. Instead, we aim to use the PLoT to describe conceptual representations and the inferences that they license across situations and domains. The process by which these inferences are implemented is not directly part of the hypothesis, though it can be very useful to consider the possible implementations when evaluating connections between the PLoT and other views of concepts Formalizing the PLoT The PLoT hypothesis as stated above is an evocative set of desiderata for a theory of concepts, more than a concrete theory itself. Indeed, it is not a priori obvious that it is possible to satisfy all these desiderata at once in a concrete computational system. We are in need of a compositional formal system a language for expressing probability

5 Concepts in a Probabilistic Language of Thought 627 distributions over complex world states. Our first clue comes from the idea of representing distributions as generative processes: a series of random steps by which the world comes to be as it is. But while generative processes are a useful way to represent probabilistic knowledge, adopting such a representation only transforms our problem into one of finding a compositional language for generative processes. The solution to this problem comes from a simple idea: if you have described a deterministic process compositionally in terms of the computation steps taken from start to end, but then inject noise at some point along the way, you get a stochastic process; this stochastic process unfolds in the original steps, except where a random choice is made. In this way a distribution over outputs is determined, not a single deterministic output, and this distribution inherits all the compositionality of the original deterministic process. The stochastic λ-calculus realizes this idea formally, by extending a universal computational system (λ-calculus) with points of primitive randomness. The probabilistic programming language Church (Goodman, Mansinghka, et al. 2008) extends the sparse mathematical system of stochastic λ-calculus into a more usable system for describing statistical processes. We now give a brief introduction to the syntax and ideas of probabilistic modeling using Church, sufficient to motivate a more formal version of the PLoT; further details and many examples can be found at Church uses a syntax inherited from the LISP family of languages (McCarthy 1960). Thus, operators precede their arguments and are written inside grouping parentheses: for example, (+ 1 2) encodes the operation add 1 and 2. We use define to assign values to symbols in our program and lambda for creating functions. We could, for example, create a function double that takes one number as an input and returns its double. The code would look like this: (define double (lambda (x) (+ x x))). Applying this function to 2 would look like (double 2) and result in the value 4. What differentiates Church from an ordinary programming language is the inclusion of random primitives. For example, the function (flip 0.5) can be interpreted as a simple coin flip with a weight (i.e., a Bernoulli random draw) outputting either true or false. Every time the function is called, a new random value is generated the coin is flipped. These random primitives can be combined just as ordinary functions are for instance (and (flip 0.5) (flip 0.5)) is the more complex process of taking the conjunction of two random Booleans. A Church program specifies not a single computation but a distribution over computations. This sampling semantics (see Goodman, Mansinghka, et al for more details) means that composition of probabilities is achieved by ordinary composition of functions, and it means that we may specify probabilistic models using all the tools of representational abstraction in a modern programming language. We will not provide a primer on the power of function abstraction and other such tools here, but we will use them in what we hope are intuitive and illustrative ways in the examples below.

6 628 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg A number of language features in Church parallel core aspects of conceptual representation. Perhaps the most familiar (and most controversial) for cognitive modeling is the use of arbitrary symbols. In Church (as in LISP) a symbol is a basic value that has only the property that it is equal to itself and not to any other symbol: (equal? bob bob) is true, while (equal? bob jim) is false. (The single quote syntax simply indicates that what follows is a symbol). Critically, symbols can be used as unique identifiers on which to hang some aspect of conceptual knowledge. For instance, they can be used to refer to functions, as when we used define to create the double function above and then reused this doubling function by name. Symbols can also be used together with functions to represent knowledge about (an unbounded set of) objects. For instance, the function (define eyecolor (lambda (x) (if (flip) blue brown))) takes a person x and randomly returns 1 an eye color (e.g., (eyecolor bob) might return blue). That is, it wraps up the knowledge about how eye color is generated independently of which person is asked about a person is simply a symbol ( bob) that is associated with another symbol ( blue) by the eyecolor function. Of course, the above representation of an object s property has a flaw: if we ask about the eye color of Bob twice, we may get different answers! Church includes an operator mem that takes a function and returns a memoized version: one that makes its random choices only once for each distinct value of the function s arguments, and thereafter, when called, returns the answer stored from that first evaluation. For instance, a memoized version of the eye color function, (define eyecolor (mem (lambda (x) (if (flip) blue brown)))) could output either blue or brown for Bob s eye color, but only one of these possibilities, to be determined the first time the function is called. This ensures that (equal? (eyecolor bob) (eyecolor bob)) is always true. Thus symbols can be used as indices to recover random properties or as labels that allow us to recover stored information. These uses are conceptually very similar, though they have different syntax, and they can be combined. For instance, we can access one function in another by its name, passing along the current objects of interest: (define eyecolor (mem (lambda (x) (if (flip 0.1) (if (flip) blue brown) (if (flip) (eyecolor (father x)) (eyecolor (mother x))))))) 1. In Church the conditional has a traditional but cryptic syntax: (if a b c) returns b if a is true, and c otherwise. Thus (if (flip) b c) randomly returns b or c.

7 Concepts in a Probabilistic Language of Thought 629 This (false, but perhaps intuitive) model of eye color asserts that the color is sometimes simply random but most of the time depends on the eye color of one of a person s parents which is accessed by calling the father or mother function from inside the eyecolor function, and so on. Symbols and symbolic reference are thus key language constructs for forming complex concepts and situations from simple ones. How does reasoning enter into this system? The fundamental operation of belief updating in probabilistic modeling is conditioning. We can define conditioning within Church via the notion of rejection sampling: if we have a distribution represented by dist (a stochastic function with no input arguments) and a predicate condition (that takes a value and returns true or false), then we can define the distribution conditioned on the predicate being true via the process: (define conditional (lambda () (define sample (dist)) (if (condition sample) sample (conditional)))) That is, we keep sampling from dist until we get a sample that satisfies condition, then we return this sample. 2 It can be cumbersome to split our knowledge and assumption in this way, so Church introduces a syntax for conditionals in the form of the query function: (query...definitions... query-expression condition-expression) Our initial distribution is the query-expression evaluated in the context of the...definitions..., and our predicate is the condition-expression evaluated in the same context. For instance, referring again to the eye color example, we could ask about Bob s mother s likely eye color, given that Bob has blue eyes: (query (define eyecolor...as above...) (eyecolor (mother bob)) (equal? blue (eyecolor bob))) Notice that there is a distinction between the definitions, which represent probabilistic knowledge reusable across many queries, and the query and condition expressions, which represent the particular question of interest at the moment. In this example, the 2. For readers familiar with Bayesian belief updating in probabilistic models, this process can be seen as taking a prior model specified by dist and generating a sample from the posterior corresponding to dist conditioned on the evidence that condition is true.

8 630 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg particular people need to be introduced only in the question of interest because the conceptual knowledge is defined over arbitrary symbols. Equipped now with a compositional formal language for representing distributions and performing conditional inference, we can revisit the probabilistic language of thought hypothesis. Within a probabilistic language like Church, knowledge is encoded in stochastic function definitions. These functions describe elements of stochastic processes that can be composed together to describe various situations, to pose various questions and to answer those questions with reasonable probabilistic guesses. Indeed, just as concepts are the stable and reusable components of human thinking, stochastic functions are the units of knowledge encoded in a Church program. Motivated from this observation, we can formulate a stronger PLoT hypothesis: Probabilistic language of thought hypothesis (formal version): Concepts are stochastic functions. Hence they represent uncertainty, compose naturally, and support probabilistic inference. Notice that this formal version realizes the informal PLoT in a precise way, showing that the hypothesis is coherent and allowing us to ask more detailed questions about plausibility. For instance we can begin to ask the usual philosophical questions about concepts of this system: What constitutes meaning? How are concepts related? How are they acquired and used? The answers to these questions can be subtle, but they are determined in principle from the basic claim that concepts are stochastic functions. For instance, on the face of it, the meaning of a stochastic function is simply its definition, and the relation between concepts is determined by constituency in the example above, the meaning of eyecolor is its definition, and it is related to other concepts only by its use of mother and father functions. However, when we consider the inferential relationships between concepts that come from conditional inference query we see additional aspects of meaning and conceptual relation. Conditioning on parentage can influence eye color, but also vice versa; conditioning on hair color could influence judgments about eye color indirectly, and so on. In the next section we give an extended example in the domain of simple team games, illustrating these foundational issues as well as exploring the empirical adequacy of the PLoT hypothesis Example: Ping-Pong in Church Consider the information shown in figure Most people conclude that TG is relatively strong, while BL is average to weak. Below we describe various patterns of evidence that we displayed to people in the guise of a Ping-Pong tournament. How can we account for people s sensitivity to the uncertain nature of the evidence in such situations? While capturing the abstract structure that remains invariant between this

9 Concepts in a Probabilistic Language of Thought 631 Figure 22.1 Screenshot of a single-player tournament. The winner of each match is indicated by a laurel wreath. particular situation and other similar situations involving different players, teams, and outcomes? This simple sports domain is built around people, teams, and games. In Church, we can use symbols as placeholders for unspecified individuals of these types. This means that we do not need to define in advance how many people participate, what the size of the teams will be, or how many games a tournament will have. Instead, we begin by describing a lexicon of abstract concept definitions useful for reasoning about these games. We define an individual player s strength (i.e., skill at Ping-Pong), personstrength, via a function that draws from a Gaussian distribution (with arbitrary M = 10 and SD = 3): (define personstrength (mem (lambda (person) (gaussian 10 3)))) Memoization ensures that the strength value assigned to a person is persistent and does not change between games. However, we assume that players are sometimes lazy in a given match. The chance of a person being lazy in a particular game is 10 percent: (define lazy (mem (lambda (person game) (flip 0.1))))

10 632 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg The overall strength of a team, in a given game, is the sum of the strength of each person on the team. If a person on the team is lazy, however, he plays with only half of his actual strength. (define teamstrength (mem (lambda (team game) (sum (map (lambda(person) team))))) (if (lazy person game) (/ (personstrength person) 2) (personstrength person))) Finally, we specify how the winner of a game is determined. We simply say that the team wins who has the greater overall strength: (define winner (mem (lambda (team1 team2 game) (if (> (teamstrength team1 game) (teamstrength team2 game)) team1 team2)))) This set of function definitions specifies a simple lexicon of concepts for reasoning about the Ping-Pong domain. The way in which we can define new concepts (e.g., teamstrength) based on previously defined concepts (personstrength and lazy) illustrates one form of compositionality in Church. The set of concept definitions refers to people (teams, etc.) without having to declare a set of possible people in advance: instead we apply generic functions to placeholder symbols that will stand for these people. That is, the concepts may be further composed with symbols and each other to describe specific situations. For instance, the inference in figure 22.1 can be described by: (query...concepts... ;The query: (personstrength TG) ;The evidence: (and (equal? team1 (winner (TG) (NG) 1)) (equal? team1 (winner (NG) (AS) 2)) (equal? team1 (winner (NG) (BL) 3)))) Here...CONCEPTS... is shorthand for the definitions introduced above a lexicon of concepts that we may use to model people s inferences about a player s strength not only in the situation depicted in figure 22.1, but in a multitude of possible situations

11 Concepts in a Probabilistic Language of Thought 633 Table 22.1 Patterns of observation for the single-player tournaments Confounded Strong indirect Weak indirect Diverse evidence evidence evidence evidence (1,2) (3,4) (5,6) (7,8) A > B * A > B A > B A > B A > B B > C B < C A > C A > B B > D B < D A > D lazy,game: B,2 B,1 B,1 C,2 Note: An additional set of four patterns was included for which the outcomes of the games were reversed. The bottom row shows the omniscient commentator s information in experiment 2. For example, in the confounded case, player B was lazy in the second game. * A > B means that A won against B with varying teams composed of several people, playing against each other with all thinkable combinations of game results in different tournament formats. This productive extension over different possible situations including different persons, different teams, and different winners of each game renders the Church implementation a powerful model for human reasoning. Gerstenberg and Goodman (2012) wanted to explore how well our simple Church model predicts the inferences people make, based on complex patterns of evidence in different situations. In Experiment 1, participants task was to estimate an individual player s strength based on the outcomes of different games in a Ping-Pong tournament. Participants were told that they would make judgments after having seen single-player and two-player tournaments. The different players in a tournament could be identified by the color of their jersey as well as their initials. In each tournament, there was a new set of players. Participants were given some basic information about the strength of the players, which described some of the modeling assumptions we made. That is, participants were told that individual players have a fixed strength that does not vary between games, and that all the players have a 10 percent chance of not playing as strongly as they can in each game. This means that even if a player is strong, he can sometimes lose against a weaker player. 3 Table 22.1 shows the patterns of evidence that were used for the single-player tournaments. Table 22.2 shows the patterns for the two-player tournaments. In all tournaments, participants were asked to judge the strength of player A. For the single-player 3. Demos of the experiments can be accessed here: _demos.html.

12 634 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg Table 22.2 Patterns of observation for the two-player tournaments Confounded with partner (9,10) Confounded with opponent (11,12) Strong indirect evidence (13,14) AB > CD AB > EF AB > EF AB > EF AC > EG BC < EF AB > GH AD > EH BD < EF Weak indirect evidence (15,16) Diverse evidence (17,18) Round robin (19,20) AB > EF AB > EF AB > CD BC > EF AC > GH AC > BD BD > EF AD > IJ AD > BC Note: An additional set of six patterns was included in which the outcomes of the games were reversed. tournaments, we used four different patterns of evidence: confounded evidence, in which A wins repeatedly against B; strong and weak indirect evidence, where A wins only one match herself, but B either continues to win or lose two games against other players; and diverse evidence, in which A wins against three different players. For each of those patterns, we also included a pattern in which the outcomes of the games were exactly reversed. For the two-player tournaments, we used six different patterns of evidence: in some situations, A was always on the same team as B (confounded with partner), and in other situations, A repeatedly played against the same player E (confounded with opponent). As in the single-player tournaments, we also had patterns with mostly indirect evidence about the strength of A by having her partner in the first game, B, either win or lose against the same opponents with different teammates (weak/strong indirect evidence). Finally, we had one pattern of diverse evidence, in which A wins with different teammates against a new set of opponents in each game, and one round robin tournament, in which A wins all her games in all possible combinations of a four-player tournament. Further experimental details can be found in appendix A. In order to directly compare the model predictions with participants judgments, we z-scored the model predictions and each individual participant s judgments. Furthermore, we reverse coded participants judgments and the model predictions for the situations in which the outcomes of the games were reversed so that both strength and weakness judgments go in the same direction. Figure 22.2 shows the mean strength estimates (gray bars) together with the model predictions (black bars) for the single- and two-player tournaments. The top

13 Concepts in a Probabilistic Language of Thought 635 A A wins Strength A loses Weakness Confounded Strong indirect Weak indirect Diverse B A wins Strength A loses Weakness Confounded partner Confounded opponent Strong indirect Weak indirect Diverse Round robin Figure 22.2 Mean strength estimates (gray bars) and model predictions (black bars) for the single-player (left) and two-player tournaments (right). The top row shows strength judgments for cases in which player A won her game(s). The bottom row shows weakness judgments for cases in which player A lost. Numbers above the bars correspond to the patterns described in tables 22.1 and Error bars indicate ±1 standard error of the mean.

14 636 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg panels display the situations in which A won her game(s). The bottom panels show the situations in which A lost. Our model predicts participants judgments in the single- and two-player tournaments very well with r = 0.98 and RMSE = A very high median correlation with individual participants judgments of r = 0.92 shows that the close fit is not merely due to an aggregation effect. These results show that our model predicts participants inferences very accurately a single, concise representation of the task is sufficient to predict people s inferences for a great diversity of patterns of evidence. A still greater variety of evidence is available by composing the basic concepts together in different ways: there is no reason for evidence not to directly refer to a player s strength, laziness, and so on. For instance: (query...concepts... ;The query: (personstrength TG) ;The evidence: (and (equal? team1 (winner (TG) (NG) 1)) (equal? team1 (winner (NG) (AS) 2)) (equal? team1 (winner (NG) (BL) 3)) (lazy NG 1))) ;additional kinds of evidence (Expt. 2) While in Experiment 1, the match results were the only source of information participants could use as a basis for their strength judgments, Experiment 2 introduced an omniscient commentator who gave direct information about specific players. After participants saw a tournament s match results, an omniscient commentator, who always told the truth, revealed that one player was lazy in a particular game. We were interested in how participants updated their beliefs about the strength of player A given this additional piece of evidence. Importantly, we do not need to change anything in the concept definitions to derive predictions for these situations, since only the way they are composed into evidence changes. Figure 22.3 shows the mean strength judgments (gray bars) together with the model predictions (black bars; see table 22.1 for the different patterns of evidence). The dark gray bars indicate participants first judgments based on the tournament information only. The light gray bars indicate participants second judgments after they received the commentator s information. The model predicts participants ratings very accurately again with r = 0.97 and RMSE = The model s median correlation with individual participants judgments is r = These results show that participants, as well as our model, have no difficulty in integrating different sources of evidence to form an overall judgment of a player s likely underlying strength. The model predicts participants judgments very accurately by being sensitive to the

15 Concepts in a Probabilistic Language of Thought A wins IR1 Strength A loses IR2 Weakness Confounded Strong indirect Weak indirect Diverse Irrelevant Figure 22.3 Mean strength estimates and model predictions. Dark gray bars = estimates after tournament information only, light gray bars = estimates after omniscient commentator info, black bars = model predictions. Error bars indicate ±1 standard error of the mean. Note: Situations IR1 and IR2 were cases in which the information given by the commentator was irrelevant; see appendix B for details. degree to which the initial strength estimate should be updated in light of new evidence provided by the commentator Intuitive Theories The examples above provide concrete illustrations of how to represent concepts as functions in a probabilistic language of thought, how a system of such concepts supports inferences that are both productive and probabilistic, and how these inferences can capture the outputs of human reasoning at a high level of quantitative accuracy. But while reasoning about team games like Ping-Pong is very illustrative, it is of relatively limited scope compared with many of the concepts involved in everyday human reasoning. In this section, we discuss how the same machinery can describe abstract concepts that are the backbone of thinking about everyday life, and that have often not fit easily into more traditional formal frameworks.

16 638 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg Intuitive theories (Gopnik and Wellman 2012; Wellman and Gelman 1992; Carey 2009), like their more familiar scientific counterparts, comprise a system of interrelated and interdefined concepts articulating a basic ontology of entities, the properties of and relations between those entities, and the causal laws that govern how these entities evolve over time and interact with each other. For instance, intuitive physics is a system for reasoning about physical objects, and intuitive psychology for reasoning about intentional agents. These are called theories because, like in scientific theories, the essential constructs of intuitive theories are typically not directly observable. Yet intuitive theories also specify how unobservable states, properties, and processes affect observable experience and thus how they support competencies such as prediction, explanation, learning, and reasoning. Intuitive theories can be found in some form in young infants, and are also to some extent shared with many other species; they are arguably the earliest and oldest abstract concepts we have (Carey 2009). They provide the scaffolding for many of children s conceptual achievements over the first few years of life. They also provide core building blocks for meaning in natural language, at the same time as they are enriched and transformed fundamentally as children develop their natural language abilities. Through adulthood, they continue to serve as the basis for our commonsense understanding of the world. Yet while these intuitive theories have long been a prime target for exploration by developmental and cognitive psychologists, linguists, and philosophers, they have not received much treatment in the literature on formal models of concepts. This may be because they do not fit well into the general-purpose mathematical and computational modeling frameworks that have been available and useful for more mundane category concepts prototypes, exemplars, and logical accounts. Starting around ten years ago, several authors began to consider Bayesian networks as formal models for intuitive theories (Tenenbaum and Griffiths 2003; Gopnik et al. 2004; Rehder 2003; Goodman et al. 2006), focusing on their causal aspects. These efforts were ultimately limited by the fact that Bayesian networks, like neural networks before them, fail to capture genuine productivity in thought. An intuitive theory of physics or psychology must be able to handle an infinite range of novel situations, differing in their specifics but not their abstract character, just as we illustrated above on a much smaller scale for an intuitive theory of tug-of-war or Ping-Pong. Hierarchical Bayesian models have been proposed as one way to increase the representational power of Bayesian networks, and they have given reasonable accounts of some aspects of abstract causal theories (e.g., Tenenbaum et al. 2011). But hierarchical Bayesian models on their own still lack sufficient representational power to address the fine-grained compositionality inherent in our intuitive theories. The PLoT allows us to take a major step forward in this regard. Both Bayesian networks and hierarchical Bayesian models of intuitive theories can be naturally written as Church programs, preserving their insights into causal reasoning and learning, but

17 Concepts in a Probabilistic Language of Thought 639 Church programs go much further in letting us capture the essential representations of common-sense physics and psychology that have defied previous attempts at formalization within the probabilistic modeling tradition. To illustrate, consider how we might capture the core concepts of an intuitive psychology a probabilistic model of how agents act rationally in response to their mental states, aiming to satisfy their desires as efficiently as possible given their beliefs. As an initial step, imagine extending the game model above to take into account the fact that laziness for a particular player in a given game may be not simply a random event, but an intentional choice on the part of a player he may estimate that the other team is so weak that it is not worth his effort to try hard. We pose this as a Church model by imagining that a player asks himself, How should I act such that my team will win? : (define lazy (mem (lambda (person game) (query (equal? (teamof person) (winner (team1of game) (team2of game) imaginedgame)))))) (define action (flip L)) action where we have helped ourselves to some helper functions to look up the team of a player and so on (and notionally extended the game into an imagined-game index). The parameter L controls the a priori tendency to be lazy; this gives a simple way of including a principle of efficiency: a tendency to avoid undue effort. The condition statement of the query specifies the player s goal for his team to win the game hypothetically assuming that this goal will be achieved. The output of the query is an action (trying hard, or not) that is a reasonable guess on the player s part for how that goal may be achieved. An inference about which team will win a match now leads to a subinference, modeling each player s choice of whether to exert his or her full effort, given the players on each team. We could further extend this model to take into account private evidence that each player might have about the strengths of the other players, expressing his or her process of belief formation about the total strengths of the two teams as an additional set of nested subinferences. The pattern of using an embedded query to capture the choices of another agent is a very general pattern for modeling intuitive psychology (Stuhlmüller and Goodman 2013). We could write down the abstract structure schematically as: (define choice (lambda (belief state goal?) (query (define action (action-prior)) action (goal? (belief state action)))))

18 640 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg where belief is taken to be the agent s summary of the world dynamics (transitions from states and actions to resulting states), and goal? is a goal predicate on states, picking out those that the agent desires. Of course, many additional refinements and additions may be needed to build an adequate model of human intuitive psychology agents form beliefs, experience emotions, and so on. Yet we hope that the ease of writing down even a simple version of a theory of mind that captures both the theory s abstractness and its potential for productive probabilistic inferences illustrates the power of the PLoT view Mental Simulation and Sampling There is a central link between the sampling semantics of Church programs, mental simulation, and the causality central to many intuitive theories. A Church program naturally expresses the causal, generative aspect of people s knowledge through the function dependencies in the program. The function dependencies dictate the causal flow of the sampling process: functions whose outputs serve as an input to another function must be evaluated first. Each run of a Church program can be interpreted as the dynamic generation of a possible world that is consistent with the causal laws as specified in the program (Chater and Oaksford 2013). Because the sampling process is stochastic, a Church program specifies a probability distribution over possible worlds, and different modes of reasoning can be seen as different forms of mental simulation on top of this basic sampling process. Although the notion of mental representation and simulation of possible worlds has had many advocates (Craik 1967; Hegarty 2004; Johnson-Laird 1983, 2006), the PLoT view integrates this idea naturally into a view of mental models that is also probabilistic, causal, and sufficiently expressive to capture core intuitive theories. We will illustrate the implications of this view via a concrete example from the domain of intuitive physics: people s judgments about the causal relations among the trajectories of physical objects in motion. We are concerned with judgments of actual causation: whether one specific event caused another specific event to happen. Consider a relatively simple scenario that consists of two billiard balls A and B, some solid walls with an opening gate, a brick, and a teleport gate that can be either active or inactive. Figure 22.4 (plate 11) shows diagrammatic illustrations of causal interactions between A and B in this world, assuming simple Newtonian elastic collisions between moving bodies. The question of interest is whether ball A s collision with ball B caused ball B to go through the red gate on the left of the screen, prevented it from going through, or did neither. The tools of probabilistic programs and sampling-based inference allow us to give a precise formal account of these causal judgments, which also accords well with intuitions of mental simulation and gives strong quantitative fits to behavioral experiments.

19 Concepts in a Probabilistic Language of Thought 641 Figure 22.4 (plate 11) Diagrammatic illustrations of four collision events in a simple physics world. Note: Solid arrows represent the actual trajectories of ball A before the collision and of ball B before and after the collision. Dashed arrows and faded circles represent the counterfactual trajectory of ball B. The brown rectangle, yellow rectangle, and blue circle represent a brick, and the entry and exit of a teleport, respectively. To explain these judgments, we first need to be able to represent the relevant physical knowledge at the right level of abstraction. Despite its simplicity, our domain already affords an infinite number of interactions between A and B, and we want a model that yields a causal judgment for each possible situation. Rather than having to specify a new model for each causal interaction of interest (as we would have to do if we adopted a Bayesian network formulation; Pearl 2000), we want to represent the general laws that govern the interactions between the objects in our world. One way of representing people s knowledge of physical object motion in Church is by writing down a probabilistic and approximate version of some aspects of Newtonian

20 642 Noah D. Goodman, Joshua B. Tenenbaum, and Tobias Gerstenberg mechanics. Functions in the Church program compute the inertial time evolution and the outcome of collisions by taking as input the mass and velocity of objects as well as more general aspects of the world such as friction and gravity. So far these are standard, deterministic simulation routines (so we leave out details). Critically, we also assume that some noise in each object s momentum is inserted just after each collision, and perhaps at other times as well, resulting in trajectories that are noisy versions of their Newtonian counterparts. Recent research has shown that people s intuitive physical judgments in several domains are well described by such noisy Newtonian simulations (Sanborn, Mansinghka, and Griffiths 2013; Battaglia, Hamrick, and Tenenbaum 2013; Smith et al. 2013; Smith and Vul 2012). Once we have a Church program that captures people s intuitive physics, we can model predictions about the future (e.g., will ball B go through the gate?) as simple forward simulations, and inferences about the past (e.g., where did ball B likely come from?) by a query of the past given the present simulating possible histories that could have led up to the current state. More subtly, a Church program can also be used to evaluate counterfactuals (e.g., would ball B have gone through the gate if the collision with A hadn t happened?). In line with Pearl (2000), the evaluation of counterfactuals in a Church program involves three steps: First, we condition all the random choices in the program based on what actually happened to estimate the unobserved values of the actual world. Second, we realize the truth of the counterfactual antecedent (e.g., that the collision did not happen) by intervening in the program execution that generated the actual world. This intervention breaks the normal flow of the program by setting some function inputs to desired values. For example, to model what would have happened if there had been no collision between A and B, we could set ball A s velocity to zero or move ball A outside of the scene shortly before the time of collision. Finally, to evaluate the truth of the counterfactual, we reevaluate all the functions downstream from the point at which we intervened in the program. This process generates a sample over counterfactual world states, and repeatedly running this process allowing for different stochastic function evaluations can be used to express people s uncertainty over what would have happened in the relevant counterfactual world. Notice that the key feature of Church that allows this process to work is that it specifies a process for sampling particular situations and makes explicit the steps of the causal history that led up to a situation (in the form of a program execution trace). Counterfactuals are then evaluated by a series of simulation steps that result in imagined counterfactual worlds. In a series of empirical studies, we have shown that people s quantitative judgments of actual causation are closely linked to such a probabilistic counterfactual analysis (Gerstenberg et al. 2012). When judging whether one event caused another event to happen, people compare what actually happened with what they think would have happened in the counterfactual world in which the candidate causal event had been absent. They appear to estimate something like the probability that the candidate cause

21 Concepts in a Probabilistic Language of Thought 643 was necessary to produce the outcome event: the probability that the outcome, which did in fact occur, would not have occurred in a counterfactual world where the candidate cause was absent. Consider the top pair of diagrams shown in figure 22.4 (plate 11). In both clips, what actually happened is identical. However, the outcome in the relevant counterfactual worlds would have been different. In figure 22.4a, ball B would have bounced off the brick if it hadn t collided with ball A. In contrast, in figure 22.4b, ball B would have gone through the gate even without the collision with ball A. As predicted by our account, people s judgments about whether A caused B to go through the gate are significantly higher for figure 22.4a than for figure 22.4b (Gerstenberg et al., 2014). In the bottom pair of cases, the contrast in the relevant counterfactual worlds was realized by comparing situations in which a teleport was either on (figure 22.4c) or off (figure 22.4d). Although people judged that A prevented B from going through the gate in figure 22.4c, they didn t think that A prevented B in figure 22.4d. In figure 22.4c, ball B would have gone through the gate via the teleport if it hadn t collided with A. In figure 22.4d, in contrast, the teleport was off, and B would not have gone through the gate even if there had been no collision with A. The fact that people s judgments differ dramatically between situations in which what actually happened was held constant supports the idea that causal and counterfactual judgments are inextricably linked. It also shows that it is not possible to give an account of people s causal judgments just in terms of what actually happened (e.g., Wolff 2007). Finally, it demonstrates the flexibility of people s intuitive theories and the critical way in which this flexibility supports mental simulation, counterfactual, and causal reasoning. Once people have learned how the teleport works, they have no trouble imagining its effects and incorporating these into their counterfactual simulations, whose outcome in turn influences their causal judgments. Although we have focused here on the example of how we can explain people s causal judgments in the domain of intuitive physics, the general framework applies equally well to any domain for which we are able to represent our knowledge in terms of a Church program. For example, we could model people s judgments about whether agent A s argument convinced agent B to try harder as a function of what actually happened and people s subjective degree of belief that B would still have tried harder had A not said anything. A Church program that captures people s intuitive understanding of psychology looks different from a Church program that captures people s intuitive understanding of physics, but we can understand people s causal judgments in terms of the same process that compares the actual outcome with the outcomes of mental simulations (sampling) of the relevant counterfactual worlds Concepts and Natural Language Thus far we have sketched a notion of concepts as stochastic functions in Church, and intuitive theories as systems of interrelated concepts. We have also described how such